孤立奇点的各种不变量及其应用

In this proposal we shall investigate intrinsic properties of certain analytic and topological invariants of isolated singularities and give proofs of several important conjectures which relate to these invariants of singularities. We want to use these results to characterize some numeric properties of intrinsic types of isolated homogeneous hypersurface singularities and as a result we shall answer a natural and important question in singularity theory and algebraic geometry: when an affine variety with an isolated singularity at origin is a cone over a nonsingular projective variety. In addition, by classifying all weighted dual graphs of simplest Gorenstein non-complete intersection singularities of dimension two, we shall answer the following question: whether there is an integral homology sphere link Gorenstein but not complete intersection singularities. The Hertling conjecture is about distribution of components which are Hodge theoretic invariants for isolated hypersurface singularities. We shall study the Hertling conjecture from two different perspectives and give certain properties of exponents which we shall introduce for an n-dimensional regular weight systems...The main parts of our research in this proposal are as follows:First,we shall solve the LWYL Conjecture for general n completely and give topological classification of simplest non-complete intersection minimally elliptic singularities.Second,we need to verify the Yau geometric conjecture and Yau number theoretic conjecture for dimension greater than five.Third,we shall also prove the Hertling conjecture is true for isolated surface singularities with modality greater than two.In addition,we shall introduce the exponents for an n-dimensional regular weight systems and prove some properties of these exponents.

本项目计划通过对孤立奇点的一些解析和拓扑不变量的本质特征的研究来证明几个关于这些不变量间内在关系的重要猜想，并运用这些结果来刻画孤立齐次超曲面奇点的内蕴数值特征，从而能解决代数几何奇点理论里面的一个重要的基本问题：在原点处有一孤立奇点的仿射簇在什么条件下为某一光滑射影簇上的仿射锥？另外本项目将通过对某些最简单非完全交奇点的拓扑型的完整分类来给出如下问题答案：是否存在具有整同调球link的Gorenstein非完全交曲面奇点？此外还将从两个不同角度研究关于一些奇点Hodge理论型不变量分布的Hertling猜想，并给出n维正则权系的指数一些基本性质的证明。..本项目主要研究内容如下：完全解决n维LWYL猜想；对最简单非完全交极小椭圆奇点的拓扑型进行完整分类；解决5维以上的Yau几何、数论猜想；对模数大于2的曲面奇点证明Hertling猜想；引入n维正则权系的指数概念并证明其有某些性质。

本项目的完全解决了LWYL猜想, 获得了齐次超曲面奇点的内蕴数值特征，从而解决了代数几何奇点理论的一个基本问题：在原点处有一孤立奇点的仿射簇在什么条件下为某一光滑射影簇上的仿射锥。这个LWYL猜想的解决有助于Zariski重数猜想的解决； 我们利用奇点解消和基本链来研究非完全交奇点，并能够给出最简单非完全交极小椭圆奇点的拓扑型完整分类，并利用其解决如下重要问题：是否存在一个具有整同调球link的Gorenstein非完全交曲面奇点；我们还找到了新的思路和方法来解决更高维的Yau几何、数论猜想从而获得齐次超曲面奇点的关于几何亏格的另一内蕴数值特征；解决了强拟凸CR流形的CR映射的刚性定理；得到了利用Kohn-Rossi上同调刻画CR流形的CR映射的非存在性定理；就CR流形引入了多重几何亏格的新概念，并用其刻画CR流形的CR映射的非存在性；解决了一定条件下的Halperin猜想、Aleksandrov猜想和Wahl猜想；对超曲面奇点引入了新的李代数并证明了这个新李代数可以完全刻画ADE奇点。还用这个李代数证明了单参数族简单椭圆奇点的Torelli型定理；得到了环曲面码的维数小于等于7的完备分类。总之在研究过程中发现了很多新方法，开始形成了自己的研究特色。